cross or vector product of unit vectors
Last time I have written about the dot product of unit vectors that is:
i.i=1
j.j=1
k.k=1
i.j=j.k=k.i=0
Today I will discuss about the cross product of unit vectors:
i x j = k
j x k=i
k x i =j
j x i= -k
k x j= -i
i x k= -j
and
i x i = j xj = k x k =0
Do you know how the above results come? If your answer is no, then let us discuss it:
I have already explained in my earlier articles that cross product or vector product between two vectors A and B is given as:
A.B = AB sin θ
where θ is the angle between A and B. A and B are magnitudes of A and B.
As i the unit vector along x axis
Therefore i x i = 1sin 0
This is because, first i is the unit vector of A along x axis and second i is the unit vector of B along x axis.
Therefore two unit vectors must be in the same direction that is x direction so the angle between them will be 0 degree. As i and i are unit vectors therefore there magnitudes will be unity. As sin 0 is 0,
Therefore above equation will become: i x i =o
Similarly
j x j =0
k x k =0
Then why i x j =k,
This is because, i along x axis and y along y axis, thus, angle between them will be 90 degree. As sin 90 = 1. As curl or rotation of two vectors give the direction of third vector
Therefore, i x j = 1 sin 90 k
i x j = k
but j x i = – k because now the direction is reversed or due to vector identity A x B is not equal to B x A.
Similarly
j x k=i and k x j = -i
k x i=j and i x k = -j
Note: I hope that now you can understand and explain everything about the cross or vector product of two unit vectors.
Reference: These articles are referred from my authored book “concepts of electromagnetic field theory” having ISBN 978-81-272-5245-8. In case of any doubt in this article or any other EMFT or physics related article, kindly post in the comment section.
What is the answer of
i . j . k = ?
And
i x j x k = ?
Please tell me the result of -jxk
dot product of adjacent vectors is zero. i.e i.j=0 then i.j.k=0…
cross product of adjacent vectors gives the next one ixj=k then ixjxk=kxk=0.
i.j.k=0
ixjxk=0
0
(i.j).k=0 as 0*k=0
ixjxk=kxk=0
Vector multication and dot product
I understand i x i = 0 and i · i = 0,
but how does i² = -1 ?
Clearly, it’s neither i x i nor i · i.
Then what is it?
Thanks for comment. Where is i square mentioned?
There, i is an imaginary number which is the root of -1. So, i*i=-1.
well the i that u are talking about is imaginary no which is root (-1) here i is aunit vector we are talking about.just same symbols used for different things
ixi=0 ; here our ” i ” is a unit vector.
i² = -1 ; here our ” i ” is an imaginary number.
You are talking about complex, not vector, i. e. i = sqrt(1)
First of all when i is an imaginary number or complex number or unreal number because i=√-1
And that’s why it is when i.i=i^2=-1
You are talking about complex, not vector, i. e. i = sqrt(-1).
As such isqr = -1
i.i=-1 as I know. but here mentioned 1 in lieu of -1.